Question

A material has mass density $\rho ,$ volume $V$, and specific heat $c.$ Which of the following is a correct expression for the heat exchange that occurs when the material's temperature changes by $\mathrm{\Delta }T$ in degrees Celsius? a) $(\rho c/V)\mathrm{\Delta }T$ b) $(\rho cV)(\mathrm{\Delta }T+273.15)$ c) $(\rho cV)/\mathrm{\Delta }T$ d) $\rho cV\mathrm{\Delta }T$

Short answer

a) (ρc / V) ΔT b) (ρcV)(ΔT+273.15) c) (ρcV) / ΔT d) ρ c V ΔT Answer: d) ρ c V ΔT

Step-by-step solution

Write down the equation for heat exchange

The formula for calculating the heat exchange, $Q$, when a material's temperature changes by $\mathrm{\Delta }T$ is: $Q=mc\mathrm{\Delta }T$

Replace mass with mass density and volume

Since mass $m$ can be written as the product of mass density ($\rho$) and volume ($V$): $m=\rho V$ Now substitute this into the heat exchange equation from Step 1: $Q=(\rho V)c\mathrm{\Delta }T$ Now, we need to find the option that matches the above expression.

Compare with the given options

Let's look at each option: a) $(\rho c/V)\mathrm{\Delta }T$ - This option divides $\rho c$ by $V$, which is not the same as our obtained expression. b) $(\rho cV)(\mathrm{\Delta }T+273.15)$ - This option adds 273.15 to the temperature change $\mathrm{\Delta }T$. This is not in our obtained expression. c) $(\rho cV)/\mathrm{\Delta }T$ - This divides $\rho cV$ by $\mathrm{\Delta }T$, while our expression has $\rho Vc\mathrm{\Delta }T$. d) $\rho cV\mathrm{\Delta }T$ - This option matches our obtained expression, which is the correct formula for the heat exchange when the temperature changes by $\mathrm{\Delta }T$. So the correct answer is: d) $\rho cV\mathrm{\Delta }T$

Key concepts

Heat Exchange

Heat exchange is a fundamental concept in thermodynamics. It describes how energy moves as heat from one system or object to another. Heat exchange occurs during processes like heat conduction, convection, and radiation. When objects come into contact, or when temperature gradients exist, heat will flow from the hotter object or region to the cooler one.
Understanding heat exchange is crucial for calculating energy transfers during heating and cooling.
- **Formula**: The formula for heat exchanged during a temperature change is given by: $$Q=mc\mathrm{\Delta }T$$ where: - $Q$: Heat exchanged (often in Joules) - $m$: Mass of the object (in kilograms) - $c$: Specific heat capacity (J/kg℃) - $\mathrm{\Delta }T$: Change in temperature (in ℃)The idea is to know how much heat is absorbed or released by the object to understand energy balance in systems.

Temperature Change

Temperature change ($\mathrm{\Delta }T$) measures how the heat exchange impacts the thermal state of a material. It highlights shifts in thermal energy, quantified by a temperature increase or decrease.
When a substance gains heat, its temperature rises. Conversely, when it loses heat, the temperature drops.
- **Role in Calculations**: The change in temperature is crucial for calculating heat exchange because it helps determine how much energy is required to achieve a certain thermal state. Given by: $$\mathrm{\Delta }T=T_{final}-T_{initial}$$ where: - $T_{final}$: Final temperature after heat exchange - $T_{initial}$: Initial temperature before heat exchange Understanding temperature changes helps predict how substances respond to energy inputs, crucial for designing thermal systems.

Mass Density

Mass density ($\rho$), often represented as $\rho$, is the mass per unit volume of a material. It's expressed in units like kg/m³ and is a measure of how tightly matter is packed in a substance.
Mass density links the material's mass to its volume and is a vital factor in calculating the mass when volume and mass density are known.- **Relation to Heat Exchange**: During heat exchange calculations, mass density helps convert volume into mass since: $$m=\rho V$$ where: - $m$: Mass (kg) - $\rho$: Mass density (kg/m³) - $V$: Volume (m³)Knowing the mass density allows you to understand the physical properties of a material, impacting how it stores and transfers energy.

Volume

Volume ($V$) is the space that a substance occupies, measured in cubic meters (m³) or liters in metric units. Volume is crucial when dealing with fluids and gases, and it affects calculations in physics and engineering.
When talking about solids, liquids, or gases, volume helps us determine the space a given mass of matter will occupy. In many thermal calculations, knowing the volume is key to linking with other properties like mass density.- **Part of Heat Exchange Formula**: In the heat exchange formula, volume helps connect the physical dimensions of a material with its heat capacity. - By knowing the volume and mass density: $$m=\rho V$$ we can calculate how much energy in heat has been exchanged when the temperature changes. Volume is essential for understanding and managing physical and chemical processes, as it influences the behavior of substances when heated or cooled.